Prove by induction the sum of complex numbers is complex number

In summary, we conducted a conversation about proving a complex number using induction. The steps in the induction were deemed correct, but the notation used was not clear and should be improved. It was also mentioned that the sum of two complex numbers is always complex.
  • #1
cbarker1
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Homework Statement
Prove by induction that for ##z## is a complex then ##\sum_{i=1}^{n}z_i##
Relevant Equations
##P(0)## is true.
If ##P(k)## is true, then ##P(k+1)## is true
See the work below:

I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.
 

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  • #2
cbarker1 said:
Homework Statement:: Prove by induction that for #z# is a complex then #\sum_{i=1}^{n}z_i#
Relevant Equations:: P(0) is true.
If P(k) is true, then P(k+1) is true

See the work below:

I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.
Hint: Enclose your code in ##, not #.

The steps in your induction are correct. I must ask, though: You know that if y = 0 that z is still a complex number, right?

-Dan
 
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  • #3
topsquark said:
Hint: Enclose your code in ##, not #.

The steps in your induction are correct. I must ask, though: You know that if y = 0 that z is still a complex number, right?

-Dan
yes. I do.
 
  • #4
The steps are all there, but your notation is bad. You are using ##z## for too many different things. You should distinguish between the sum of n versus the sum of n+1. Call the sum of n ##s_n## and the sum of n+1 ##s_{n+1}##.
Also, it would be a little more clear if, before the last line, you said that ##s_n + z_{n+1}## is the sum of two complex numbers so it is complex.
 
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  • #5
In general:

If $$a, b \in S \implies a + b \in S$$ then:$$a_1, \dots a_n \in S \implies \sum_{i=1}^{n}a_i \in S$$
 
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1. What is induction in mathematics?

Induction is a mathematical proof technique used to prove statements about a set of numbers or objects. It involves proving that a statement is true for a base case, and then showing that if the statement is true for any given case, it must also be true for the next case.

2. How do you prove a statement using induction?

To prove a statement using induction, you must first show that it is true for a base case, usually the smallest or simplest case. Then, you assume that the statement is true for any given case, and use this assumption to prove that it must also be true for the next case. This creates a chain of reasoning that proves the statement is true for all cases.

3. What is the base case in a proof by induction?

The base case is the starting point for a proof by induction. It is usually the simplest or smallest case that can be proven to be true. For example, in a proof that the sum of the first n natural numbers is n(n+1)/2, the base case would be n=1, as 1 is the smallest natural number.

4. How does induction apply to complex numbers?

Induction can be used to prove statements about complex numbers, just as it can be used for real numbers. In the case of proving that the sum of complex numbers is a complex number, the base case would be when n=1, and the statement would be true for all positive integers n.

5. Can induction be used to prove all statements about complex numbers?

No, induction can only be used to prove statements that can be broken down into a series of smaller cases. There are some statements about complex numbers that cannot be proven using induction, and may require other proof techniques such as contradiction or direct proof.

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